To date, several studies have provided evidence on increasing educational inequalities due to the first school closures in spring 2020. However, only a still limited number of studies have directly addressed socioeconomic and/or ethnic differences in learning loss. A recent systematic review on the effects of school closures on student achievement by Hammerstein et al. (2021) included in total eleven relevant studies (published till April 30, 2021), whereas only seven studies also reported results regarding socioeconomic and/or ethnic disparities. A more inclusive review (Helm et al., 2021b), also considering conference presentations and gray literature, identified twelve additional studies on educational inequalities. Notably, eleven of these 19 studies are from the United Kingdom and United States. In contrast to this body of research directly addressing achievement disparities, a considerably larger proportion of studies has indirectly addressed the issue by dealing with socioeconomic and/or ethnic differences in distance learning (time spent in learning, available IT-equipment, etc.), that might in turn, contribute to increasing educational inequalities [Andrew et al., 2020; Grewenig et al., 2020; Reimer et al., 2021; for a review of studies from Germany, Austria and Switzerland see Helm et al. (2021a)].

Using biannually collected achievement data of a sample of about 350,000 Dutch primary students (aged 8–11 years), Engzell et al. (2021) directly addressed increasing inequalities. The first test took place in January and February, thus in 2020 directly before the school closures in March, and the second test took place in June, at the end of the school year. The authors compared the learning gains during this period in 2020 with the learning gains during the same period in previous years and found that in 2020 students’ learning progress–estimated using an achievement composite score covering mathematics, reading, and spelling–was about 0.08 SD lower than in previous years. Using various methods to control for confounders (e.g., propensity score weighting), they show that students with less educated parents (i.e., none of the parents has a degree above lower secondary education) experienced a significantly larger learning loss (≈0.1 SD) than their peers with higher educated parents (learning loss ≈0.075 SD). Notably, they did not find any differences in learning loss between age groups, and they did not test whether learning loss differs with the ethnic background of the students. In a Belgian study, Maldonado and De Witte (2020) compared achievement data from 6th graders assessed at the end of the school year 2020 with respective data from the previous years. By linking aggregated data on the school level over time (i.e., a panel at school level not at individual level) and controlling for various school-level characteristics, they found that achievement differences in 2020, both within schools and between schools, were larger than in previous years. Moreover, they report that learning loss was more pronounced in schools with a higher share of students with low SES (determined by the mothers’ educational level and students receiving financial support for schooling). However, they did not find an association between learning loss and the share of students who do not speak the instruction language at home. Note that this study used aggregated school-level data and thus inferring on the individual level is not warranted (ecological fallacy). Providing more support for increasing disparities, Pier et al. (2021) used formative assessment data from 18 Californian districts (about 50,000 students) to compare language and mathematics test score gains within the school year 2019/20 with test score gains within the previous year 2018/19. They found a larger learning loss for socioeconomically disadvantaged students and English language learners.

However, it should be mentioned that there are also some studies that do not find any new educational inequality due to coronavirus (for an overview see Helm et al., 2021b). Interestingly, whereas Helm et al. (2021b) show in their review that studies from English speaking countries (United States, United Kingdom, Australia), the Netherlands and Belgium very clearly confirm increased inequalities, none of the three studies from German speaking countries found support for growing disparities. For example, two German studies (Depping et al., 2021; Schult et al., 2021) compared student (grades 4 and 5) tests scores in reading and mathematics assessed in fall 2020 with test results from pre-corona cohorts and did not find evidence that disparities increased between schools with a disadvantaged student body (low SES, migration background) and schools attended by a lower share of disadvantaged students.

Beside these studies that provide direct support for the hypothesized increase in disparities, there are a growing number of studies that indirectly addressed COVID-induced inequalities by focusing on various aspects of learning during school closures that may account for a widening achievement gap between students with different family backgrounds.

Several surveys addressed the question of whether low SES students became less involved in learning during school closures than socioeconomically more advantaged students. The results of a German study by Dietrich et al. (2021) show that socioeconomically disadvantaged students spent less time on learning during school closures than their better off peers. Similar results are reported in a study from Spain by Bonal and González (2020) who show SES-differences in an opportunity to learn measure, covering the learning time, frequency of online lessons, and teacher contact during school closures. Also considering pre-lockdown differences in learning time, Andrew et al. (2020) and Grewenig et al. (2020) analyzed whether SES differences in learning time increased during lockdown. For England, Andrew et al. (2020) reported an increasing SES gap in learning time for primary school students, but not for secondary school students. Not differentiating between primary and secondary schools, Grewenig et al. (2020) did not support the increasing SES differences in learning time in a German study.

Using data from a reading app, a Danish study by Reimer et al. (2021) report that the time 4th and 5th graders spent using the reading app changed with the school closures. Notably, differences in reading time increased during the first lockdown phase (i.e., till the Easter holidays) with socioeconomically advantaged peers showing a steeper increase in reading time. However, no differences in the second phase (after the Easter holidays) were found, where students attended school on a limited schedule.

Various studies have also shown that socioeconomically disadvantaged students have limited access to learning resources at home (e.g., own study space, available computer, or tablet) as well as to those provided by schools [see for example the review on surveys in Germany, Austria, and Switzerland by Helm et al. (2021a)].

Moreover, several studies have examined whether there are socioeconomic or ethnic differences in the extent of parental support and involvement in distance learning. Bonal and González (2020) found no SES differences in parental support (whether parents helped children with school tasks during lockdown) among primary school children and upper secondary school children. However, for lower secondary school children, higher SES parents were more likely to report helping their children with school tasks than lower-SES parents. A German study by Sander et al. (2021) focusing on SES-differences in involvement found that higher-SES parents paid more attention to the establishment of structures during distance learning (e.g., regular study times). Interestingly, they found that lower-SES parents and non-German speaking parents reported more process-focused learning support (e.g., help to apply meaningful learning methods). Similarly, a study from Portugal (Ribeiro, et al., 2021) also found that lower-SES parents were more involved in terms of time in their children’s learning during school closures.

To sum up, there is growing evidence that socioeconomically disadvantaged and ethnic minority students experienced a larger learning loss during school closures than their better off peers and ethnic majority peers, respectively. Moreover, disadvantaged students had restricted access to learning resources available at home and to those provided by schools and spent less time on learning during school closures. However, it is still unclear whether or not–and to what extent–these aspects of learning during school closures (e.g., time spent on learning) contributed to increased disparities in achievement. Moreover, little is also known about other factors of the learning context at home that may account for an increased learning gap. Here, parental involvement in the distance learning of their children might be of special importance (Education Endowment Foundation, 2020), as missing instruction due to school closures had to be at least partly compensated by parents. This is especially true for younger children whose self-regulatory skills are still developing and who need more support and guidance for successful learning (Vandevelde et al., 2015; Cottingham, 2020). Several studies that have considered parental involvement in distance learning till date have largely assessed involvement only through few items focusing either on the quantity of involvement (e.g., Grewenig et al., 2020; Nusser, 2021; Ribeiro et al., 2021) or on the general provision of learning support during school closures (e.g., Bonal and González, 2020). Moreover, these studies provide mixed results on socioeconomic and ethnic differences in parental involvement. Therefore, a broader consideration of the concept of parental involvement seems to be purposeful.

Parental childrearing practices in general (Pinquart, 2016) and especially parental involvement in schooling (Barger et al., 2019) are considered to affect children’s academic achievement. Following Grolnick and Slowiaczek (1994), parental involvement in schooling is defined as the dedication of resources (e.g., time, energy, money) to the child’s academic lives. It encompasses two broad forms: school-based involvement and home-based involvement (Barger et al., 2019). School-based involvement comprises–amongst others–parents’ direct contacts with school. Home-based involvement is multifaceted, covering activities such as talking with children about school, encouraging them in their academic efforts, and helping children with homework (Pomerantz et al., 2007; Pomerantz and Grolnick, 2017). Primary school teachers’ distance instruction during school closures was largely characterized by assigning homework-like tasks to students, which they had to complete at home (Grewenig et al., 2020; Weber et al., 2021). Thus, it is reasonable to put a special focus on parental involvement in distance learning, which is conceptually equivalent to homework involvement.

However, whereas research indicates that overall parental involvement in schooling is conducive for children’s academic development (Barger et al., 2019), research on the effects of parental homework involvement is ambiguous. A meta-analysis of Patall et al. (2008) showed small positive correlations between homework involvement (helping, monitoring, etc.) and achievement in the primary school and high school years, but a negative correlation for middle school students. In contrast, the results of a recent meta-analysis that considered a larger body of studies indicated that homework involvement is negatively associated with academic outcomes irrespective of the children’s developmental stage (Barger et al., 2019).

Distinguishing between quantity and quality of homework involvement appears to resolve some inconsistencies (Dumont et al., 2014; Moroni et al., 2015; Pomerantz and Grolnick, 2017). This body of research largely draws on self-determination theory (SDT; Grolnick et al., 1997; Grolnick, 2003, 2009). In a nutshell, SDT posits that children are born with three psychological needs which are facilitated by three parenting dimensions. By allowing children choices, supporting their initiative, and taking children’s perspective (i.e., autonomy support), parents can support the need to feel autonomous. In contrast, controlling parenting–i.e., the conceptual opposite of autonomy support–is restricting children’s choices and thus, frustrates the need for autonomy. Communicating clear rules and expectations–i.e., providing structure–facilitates the need for competence. Finally, the need to feel related to others is satisfied by caring, supportive, and positively involved parenting. Following SDT, the fulfillment of these needs fosters motivational resources necessary for developing competence (Grolnick, 2003, pp. 12–17, 2009, p. 165; Pomerantz and Grolnick, 2017). Applying this theoretical framework to the study of parental homework involvement, frequent assistance and helping with homework (i.e., quantity) is argued to be negatively associated with child achievement, because frequent helping inhibits children’s autonomous motivation and feelings of competence and thus interferes with academic development (see also Silinskas et al., 2015; Pomerantz and Grolnick, 2017). In this vein, Silinskas et al. (2015) found that, controlling for achievement in grade 1, frequent helping with homework was related to lower mathematic achievement in grade 4. Similar results have been reported by Moroni et al. (2015). Moreover, several studies show that forms of parental homework involvement (i.e., quality) that are controlling and characterized by negative parental affect have harmful effects on children’s academic development (Dumont et al., 2012; Moroni et al., 2015; Silinskas and Kikas, 2019). For example, Moroni et al. (2015) found that controlling involvement (e.g., parents interfere when child is doing her/his homework) predicted lower reading achievement in grade 5, and Dumont et al. (2012) reported that parent-child-conflicts during homework–indicating negatively affective involvement–predicted German and mathematics grades in 8th graders. Both studies have controlled for prior achievement. In contrast, parental homework involvement that is autonomy supportive, responsive to children’s needs, provides structure and is characterized by positive affect is conducive to children’s academic development (Grolnick et al., 2015; Moroni et al., 2015; Grolnick, 2016; Pomerantz and Grolnick, 2017; Silinskas and Kikas, 2019). For example, Grolnick et al. (2015) report results of a path analysis that show that autonomy supportive involvement (e.g., encouraging children to find solutions on their own) and structuring involvement (e.g., clear and consistent rules and expectations) predict English grades in 7th graders, even when controlling for grades of the previous year.

There has long been the hypothesis that the effects of socioeconomic status on various aspects of child development are mediated by parenting (Bradley and Corwyn, 2002; Grolnick, 2003; Conger and Donnellan, 2007). Amongst others, it is argued that parents with lower SES are more likely to experience economic stress, which in turn makes them more emotionally distressed and distracts their attention from childrearing (e.g., being less involved). Further, low SES parents invest less resources in their offsprings’ development due to a lack of available resources (money, knowledge, etc.) or SES-specific values (Kohn, 1959; Bradley and Corwyn, 2002; Conger and Donnellan, 2007). Similarly, the ethnic background of parents might be associated with different values that affect parenting and language barriers of immigrant parents can complicate involvement (Antony-Newman, 2019). In this context, Fleischmann and de Haas (2016) found in a Study from the Netherlands that Turkish and Moroccan parents more strongly value obedience in their children than their Dutch counterparts. However, ethnic differences in parental school involvement (Turkish and Moroccan parents reported lower levels of school involvement) were entirely explained by differences in parental education and language proficiency. Ethnic differences in parenting goals were not associated with differences in involvement. In the context of this study, low parental proficiency in the language of instruction could simply pose a challenge for helping students with their homework.

Regarding children’s academic development, parental homework involvement is a candidate mediator of family background effects. In testing this hypothesis, Dumont et al. (2012) found no significant indirect effects of family background variables (SES-indicators and migration background) on achievement. Thus, mediation was not supported. However, SES-indicators and immigration status were associated with various aspects of homework involvement. Specifically, immigrant students reported that their parents are less supportive and less competent in helping them with homework, but also less controlling than parents from non-immigrant students. Low-SES students reported less support, less competence, and more homework-related conflicts than their socioeconomically more advantaged peers. Similar results regarding the association between SES and homework involvement are reported in various other studies (Silinskas et al., 2010; Moroni et al., 2014; Sander et al., 2021). For example, Moroni et al. (2014) found that higher SES was associated with more autonomy supportive involvement. Moreover, immigrant students reported less frequent parental involvement (quantity) and less autonomy supportive involvement. In contrast to Dumont et al. (2012), migrant students characterized their parents as more controlling than non-migrant students. Thus, there is evidence that higher-SES parents use more conducive and less detrimental forms of homework involvement. For parents with a migration background, research is somewhat inconclusive.

As outlined above, social and ethnic disparities in school achievement are expected to have grown during school closures. However, although there is growing research confirming the expected learning losses and growth of disparities, the number of studies is still small and largely limited to English-speaking countries (see Helm et al., 2021b; Hammerstein et al., 2021). Thus, our first research question (RQ 1) is:

In line with the conclusion of recent reviews (Helm et al., 2021b; Hammerstein et al., 2021), we hypothesize that social and ethnic disparities have grown during school closures. Notably, whereas recent studies have analyzed achievement gaps between cohorts (e.g., comparing disparities in 6th graders of school year 2019/2020 with disparities of 6th graders from previous years), this study takes a within-subject perspective and analyzes whether social and ethnic disparities have grown during school closures compared to the pre-lockdown period of the school year 2019/2020.

Moreover, a plethora of studies has focused on the association between social and ethnic family background and variables such as learning time, parental involvement, and access to learning resources, that may account for growing disparities (Andrew et al., 2020; Bonal and González, 2020; Grewenig et al., 2020; Helm et al., 2021a; Dietrich et al., 2021; Reimer et al., 2021; Sander et al., 2021). However, all these studies are not based on achievement data and thus, do not directly address gaps in achievement.

From a mediation analytical perspective (e.g., MacKinnon, 2008, see also Figure 1), studies have tested c-paths (i.e., the effects of family background on learning loss without considering mediators) and a-paths (i.e., the effects of family background on mediators). To the best of our knowledge, however, no study has yet tested a mediation model to explore the reasons for grown disparities due to school closures. Using a panel sample of Austrian second graders, the current study focuses on a whole mediation process (see also Figure 1) by taking up parental involvement in distance education–conceptually equivalent to homework involvement–as mediators. In detail, we investigate the following research question (RQ2):

Taking an SDT-perspective on parental homework involvement (Grolnick, 2016; Pomerantz and Grolnick, 2017), we hypothesize that family background is associated with homework involvement in such a way that lower-SES parents show less conducive (e.g., autonomy supportive, structuring, positively affective) and more detrimental (e.g., controlling, negatively affective) forms of homework involvement (a-paths). As research on ethnic family background is inconclusive, we do not formulate a directional hypothesis. Socioeconomic and ethnic differences in parental involvement in distance learning might be due to differences in values and stress exposure that already existed before the coronavirus crisis, but which gain in importance due to the shift in learning from the classroom to the home. Put simply, it makes a difference if a parent tends to be controlling due to specific values or stress exposure and thus, gets involved in homework, in a controlling way, some days a week for up to an hour, or if she/he gets involved, in a controlling way, 5 days a week for several hours. Moreover, a recent research review indicates that lockdowns lead to more parental stress and stress outcomes (e.g., depression, burnout), especially in low SES families and in parents of younger children, which was also accompanied by harsher and less warm parenting (Proulx et al., 2021). Thus, lockdowns might especially increase stress exposure of low SES parents, which in turn affects their involvement in distance learning.

Following research on parental homework involvement (Dumont et al., 2014; Silinskas et al., 2015), we argue that involvement in distance learning is associated with reading achievement (b-paths). Specifically, we hypothesize that autonomy supportive, structuring, and positively affective forms of parental involvement in distance learning are conducive to the development of reading achievement during school closures, whereas controlling and negatively affective involvement is detrimental (Pomerantz and Grolnick, 2017). Notably, whereas research on homework involvement indicates that more involvement is associated with lower achievement (Barger et al., 2019), this might not be the case for parental involvement in distance learning. Due to canceled teacher instruction in classrooms, children would have been left without direct adult support and guidance that largely had to be provided by parents. Thus, we argue that especially young children needed the support, guidance, and involvement of their parents on a largely daily basis (Vandevelde et al., 2015; Cottingham, 2020; Education Endowment Foundation, 2020).

Finally, we hypothesize that family background effects on reading achievement are at least partly mediated by parental involvement in distance learning. Thus, we expect significant indirect effects of background variables on reading, but background effects might still be significant after controlling for parental involvement (i.e., c’-paths). Significant c’-paths might be due to other factors such as the quality of distance teaching and the quality of the study space at home, that might differ between high-SES and low-SES schools (Andrew et al., 2020; Bonal and González, 2020), or SES-differences in child skills (e.g., self-regulation; Bradley and Corwyn, 2002) that affect learning during school closures.

This research partly draws on data from an ongoing study on the use of learning progress assessment (LPA) in primary school. The W3-study (W3 stands for “Wir Wollen’s Wissen!” in English “We want to know how our students’ skills are doing!”; Weber et al., 2020) started in school year 2018–2019. In fall 2018, schools were informed about the project via the Upper Austrian educational administrations. A total of 28 schools with 48 classes decided to participate in the study. Parents of 745 students gave their written consent to participate in the study. It was planned that 1st grade students of the 2018/19 school year would participate in the project over the entire span of primary school, which lasts 4 years in Austria. Some teachers, however, decided to leave the project after the first year. Thirty-five classes with 579 2nd graders remained in the project in 2019/20. Initially, it was planned that the students should complete eight short internet-based reading comprehension LPA tests (Souvignier et al., 2021) at fixed intervals of 3 weeks throughout the school year. The LPA tests are completed independently by students during self-study periods or group tests at school, depending on the number of available computers. It has been shown that students can complete the LPA test independently as early as grade 1 (Salaschek and Souvignier, 2013). Moreover, the students of our sample already had 1 year of experience in using the LPA tests. Four LPA tests between March and the end of June were largely not completed due to school closures, which started in Austria on the 16th March and ended on the 18th May. For example, the eighth LPA-test (test period during the first 3 weeks of June) was only used in eleven classes. Notably, although the whole predefined testing period of the eighth and last LPA-test was after school closures, a broader use was hampered by shift-schooling (see later) and three additional days off from school within this period. Moreover, the specific situation after returning to schools must be considered that kept teachers from LPA-testing. Thus, to ensure a broader post-lockdown reading assessment, teachers were asked to administer a standardized reading comprehension test (ELFE II; Lenhard et al., 2017) at the end of the school year (late June/early July). The ELFE II was chosen because it is conceptionally equivalent to the LPA-tests (see also Reading Achievement section) and has the advantage that it could be simultaneously administered within 30 min to the whole class. Although there is a computer version of the ELFE II, we choose to use the print version of the test, since most classes have only a few computers that could be used for testing. The test was administered by the teachers to their students, using standard instructions. Moreover, we also conducted a parent survey on different aspects of distance learning (including parental involvement). Twenty-five teachers agreed to administer the ELFE-test and we received parent questionnaires from 21 classes. A parent survey covering family background variables (including SES, ethnic background) was already conducted in the school year 2018/19. For the current study, we used data from 25 classes that regularly used the LPA tests before the school closures and took the ELFE-test after lockdown (n = 409 students; 50.1% females). The mean age of the students at the beginning of the school year was M = 7.7 years (SD = 0.56). 14.8% of the students have a migration background, which is significantly below the rate for the Upper Austrian primary school population [23%; χ²(1) = 10.937, p < 0.05]. In about one-third (29.7%) of the families, at least one parent has a university degree. A further 22.5% have a university entrance qualification. Parental education in the sample is representative of the general parent population of Upper Austrian primary school students [χ²(3) = 5.262, p > 0.05].

For the current study we used three time points (see Figure 2). For t1 and t2, LPA reading data are available and for t3, we used data from the ELFE II test. LPA tests at t1 were carried out in November 2019 and LPA tests at t2 between the 24th February and the March 6, 2020. ELFE II tests were administered between the 22nd June and the July 11, 2020. The interval between t1 and t2 covers approximately 16 weeks (including 2 weeks Christmas holidays and 1 week semester break), thus, students regularly attended school for about 65 days (i.e., 13 weeks) in this period. The interval between t2 and t3 covers roughly 17 weeks, whereas schools have been closed in Austria from the 16th March to the 18th May (including one-and-a-half-weeks for the Easter holidays). To reduce class sizes after reopening schools, students attended classes only half the week (shift-schooling). One group of students had to learn at home, while the other group was attending classes. Taken together, students roughly attend classes between t2 and t3 for 25 days, that is, approximately 60 days less than normal.

We applied structural equation modeling (SEM) using Mplus 8 (Muthén and Muthén, 1998–2017). To answer RQ1, we estimated the two models depicted in Figure 3: A pre-lockdown model and a lockdown model. Specifically, we regressed reading achievement at t2 and t3, respectively, on socioeconomic and ethnic family background variables and reading achievement assessed at the preceding time point t–1 (i.e., reading t2 was regressed on reading t1 and reading t3 was regressed on reading t2). Reading achievement was modeled as latent variables using the three subtests as indicators. Significant effects of family background variables would indicate that among students who started at t–1 with the same level of reading achievement, family background is associated with their achievement at the end of the respective period (pre-lockdown or lockdown). Looked at another way, family background effects imply that social and ethnic disparities at t2 and t3, respectively, are bigger than expected due to differences at t–1, and thus indicate a relative growth in the achievement gap. Note that due to different reading tests we cannot conclude whether disparities increase on a common scale.

To answer RQ2, we have extended the lockdown model to the latent mediation model shown in Figure 4. The single item measures of parental involvement were included as manifest variables. The mediation hypothesis would be supported if indirect effects are statistically significant (Cheong and MacKinnon, 2012). Due to the high number of parental involvement measures, we estimated separate models for each mediator. In doing so, we end up with seven mediation models. Finally, we included sex of child and learning time as a control variable in the mediation models.

Moreover, as the two ethnic background measures were highly correlated (r = 0.68, p < 0.001), we estimated separate models (for RQ1 and RQ2) for non-German language use and migration background.

To evaluate the fit of the SEMs, we used the cut-offs proposed by Schermelleh-Engel et al. (2003). A good fit is indicated by χ²/df ≤ 2, CFI ≥ 0.97, RMSEA ≤ 0.05, left boundary of the 90% confidence interval (CI) of the RMSEA equals 0 and SRMR ≤ 0.05. An acceptable fit is indicated by χ²/df ≤ 3, CFI ≥ 0.95, RMSEA ≤ 0.08, 90% CI close to the RMSEA and SRMR ≤ 0.10.

Given the multilevel structure of the data, we used TYPE = COMPLEX in Mplus which applies a sandwich estimator that adjusts for biased standard errors due to clustering (students clustered in classes), provided that the sample size at cluster level is at least 25 (Huang, 2018).

The rate of missing data ranged from 1% (sex) to roughly 10% in the reading tests and about one-third for the parent reports. Whereas missing data on the reading tests were due to students’ absence (e.g., illness) during the test period, missing data on parent reports are mainly due to unit nonresponse (i.e., parents did not fill out the questionnaire). As indicated by Little’s MCAR-test (Little, 1988), the variable means significantly differed between missing data patterns [χ²(1683) = 2004.28, p < 0.001]. For example, children with missing family background variables (HISEI, parental education, migration background, etc.) scored significantly lower on the ELFE II test. Thus, the results suggest that missing data were the consequence of a missing at random (MAR; see Enders, 2010) mechanism, i.e., the missingness depends on other study variables. To appropriately deal with the missing data, we used a full information maximum likelihood (FIML) estimation, which is an appropriate treatment of missing data under the MAR mechanism (Rioux and Little, 2021).

Finally, as especially LPA-scores were non-normal (skewness ranged from 0.63 to 2.02 and kurtosis from 0.011 to 7.05) and we used ordinal parental involvement indicators, the models were estimated using a robust maximum likelihood estimation (MLR), with standard errors robust to the non-normality of observations and to the use of ordinal variables (Finney and DiStefano, 2006).

Table 2 shows the results regarding RQ1. Fit indices indicate an acceptable fit. The columns labeled section A provide the results for the pre-lockdown and lockdown period using language use as an indicator of the ethnic background. The columns labeled section B present the results using migration background as predictor. Both models for the pre-lockdown period show that reading at t2 is associated only with reading at t1 (Section A: β = 0.703, p < 0.001; Section B: β = 0.700, p < 0.001). However, in the models for the lockdown period we found significant effects of SES (Section A: β = 0.122, p < 0.05; Section B: β = 0.189, p < 0.01) and non-German language use (Section A: β = −0.274, p < 0.001). The effect of migration background is not significant.

To statistically compare the effects of the pre- and peri-lockdown period, we estimated confidence intervals (CI) for the differences in β-coefficients (i.e., β_SES,Lockdown–β_{SES,Pre-lockdown}, etc.) applying the method proposed by Zou (2007) ² and implemented in the cocor R-package (Diedenhofen and Musch, 2015). Moreover, we report the effect size Cohen’s q (Cohen, 1988) to quantify differences in β-coefficients (q < 0.1 = no effect, 0.1 ≤ q < 0.3 = small effect; 0.3 ≤ q < 0.5 = moderate effect; q ≥ 0.5 = large effect). These results are also reported in Table 2. The CIs for the difference in the SES-effects in section B [Pre-Lockdown vs. Lockdown: q = 0.146, 95%-CI (0.056, 0.231)] and the difference in the effects of non-German language use in section A [Pre-Lockdown vs. Lockdown: q = −0.309, 95%-CI (−0.387, −0.215)] do not contain 0, thus, indicating a significantly stronger effect of SES and non-German language use during the lockdown period than during the pre-lockdown period. However, SES-effects in section A do not differ significantly [Pre-Lockdown vs. Lockdown: q = 0.072, 95%-CI (−0.018, 0.160)].

To sum up, reading scores are associated with family background at t1, t2, and t3 (see Table 1). For the pre-lockdown period we found no effects of the family background variables on reading comprehension. Thus, social and ethnic disparities at t2 are not larger than expected due to preexisting disparities at t1. However, the results for the lockdown period indicate that social and ethnic disparities after the lockdown (t3) are larger than expected due to pre-lockdown disparities (t2).

Results regarding RQ2 are shown in Figure 4. Note that only significant coefficients for a, b and c’-paths are displayed. Detailed results are reported in Supplementary Tables A3, A4 in the supplement. Note that in the case of two reported values separated by a slash, the first value refers to the results using language use as indicator of the ethnic background (Section A in Supplementary Table A4) and the second value refers to results for migration background (Section B in Supplementary Table A4). Fit indices indicate an acceptable to good fit (χ²/df = 1.78–2.37, RMSEA = 0.044–0.058, CFI = 0.966–0.975; TLI = 0.942–0.964, SRMR = 0.032–0.047).

First, there is only scant support for the hypothesis that family background is associated with parental involvement in distance learning (a-paths). Specifically, high-SES parents more frequently reported that they established structures for distance learning (β = 0.186, p < 0.05) and high-SES parents also reported less intrusive involvement (β = −0.170/−0.178, ps < 0.01). Moreover, parents with a migration background reported higher levels of controlling involvement (β = 0.168, p < 0.01). Second, we hardly found any support for the hypothesis that parental involvement is associated with reading achievement at t3 (b-paths). Only one out of 14 analyses resulted in a significant effect of parental involvement on reading. Higher levels of autonomy supportive involvement were associated with better reading at t3 (β = 0.118, p < 0.05). After controlling for multiple testing (Bonferroni-Holm), even this association becomes insignificant. Third, because the lack of significant a-paths and b-paths direct effects of family background (c’-paths) remain largely statistically significant and hardly change in size (SES: βs = 0.108–0.120/0.178–0.188, ps < 0.05–0.10/.01–0.05; LU: βs = −0.273 to −0.294, ps < 0.001) and thus, we did not find any significant indirect effects of family background on reading achievement.

Finally, sex of child was associated with controlling involvement and intrusive involvement. Parents were more involved in a controlling (β = −0.208/−0.204, p < 0.001) and intrusive (β = −0.093/−0.089, p < 0.05) way in distance learning of boys than that of girls. Moreover, learning time was positively associated with establishing structures (β = 0.084/0.097, p < 0.10/0.05) and positive interpersonal involvement (β = 0.121/0.127, ps < 0.05) and reading at t2 was negatively related to intrusive involvement (β = −0.111/−0.115, ps < 0.05).